{"id":"526d1feb-2489-487d-b225-548e437eb4ee","arxiv_id":"2607.13186","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The q-Cayley determinant of q-right-quantum matrices equals a Valiant-style clow determinant and is computable by a polynomial-size algebraic branching program.","lead":"This paper constructs a polynomial-size algebraic branching program that computes the quantum determinant of right-quantum matrices. It claims to be the first efficient exact algorithm for a noncommutative determinant in the quantum-group setting, and also makes the permanent exactly computable for a related special class.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"In Lemma 9.6 the auxiliary matrix Q is not generally q-RQ when the q-parameters differ; the reduction to Cdet_q(Q)=0 is invalid for multiparameter q.","rationale":"The reader's verdict was already CONDITIONAL and correctly located the type-3 collective cancellation in Lemmas 8.18/9.6 as the most delicate part. My stress test identifies a more specific defect within Lemma 9.6: the auxiliary matrix Q obtained by duplicating a column is not a q-RQ matrix for multiparameter q, because the defining relations depend on the label of the duplicated index and the inherited relations for A force a different q-parameter than the one Q's position demands. This is separate from the reader's noted Section 9.1 swap issue and from the q-weight matching question; it directly breaks the reduction of a type-3 signed sum to the vanishing of a determinant with equal columns. The one-parameter case q_{ij}=q and the q=1 RQ case escape this particular problem, so the concern is specific to the full multiparameter theorem. A concrete rewriting-system computation settles it. Since the proof may be repairable by replacing Q with a properly q-shifted column operation or by proving the cancellation directly, the appropriate verdict remains conditional rather than reject; no change from the reader's assessment.","tokens_in":28784,"tokens_out":25582,"duration_ms":199504,"concrete_test":"Let n=3, k=2, q_{12}=u, q_{13}=v, q_{23}=w with v≠w. Form Q from a generic q-RQ matrix A by Q_{1r}=Q_{2r}=a_{2r}, Q_{3r}=a_{3r}. Compute the normal form of D = Q_{1,2}Q_{3,1} - u Q_{1,1}Q_{3,2} - v u Q_{3,1}Q_{1,2} + v Q_{3,2}Q_{1,1} modulo the quadratic ideal (3.2) on the variables of A, using a Gröbner basis / rewriting system. If the normal form is nonzero (it is a nonzero multiple of (v-w)(u a_{3,1}a_{2,2}-a_{3,2}a_{2,1})), then Q violates relation (3.2), disproving the claim Q∈B^q_n and invalidating Lemma 9.6 as written.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central equality Theorem 4.1 is proved via type-3 collective cancellation. Lemma 9.6 defines Q by Q_{1r}=a_{kr} and Q_{cr}=a_{cr} for c≥2, and asserts Q∈B^q_n because its columns are a subcollection of A's columns. This fails for multiparameter q. Take n=3, k=2, q_{12}=u, q_{13}=v, q_{23}=w with v≠w. Then Q has first two indices 1 and 2 both equal to A's index 2. Relation (3.2) for Q with k=1, ℓ=3, i=1, j=2 reads Q_{1,2}Q_{3,1}-u Q_{1,1}Q_{3,2} = v u Q_{3,1}Q_{1,2} - v Q_{3,2}Q_{1,1}. But the left side is exactly the left side of relation (3.2) for A with k=2, ℓ=3, i=1, j=2, whose right side has w in place of v. Unless v=w, the two relations force (v-w)(u a_{3,1}a_{2,2}-a_{3,2}a_{2,1})=0 in B^q_n, which is not an identity for generic variables. So Q is not generally a q-RQ matrix. Consequently the applications of Theorem 9.1 and Lemma 3.1 to Q in Lemma 9.6 are unjustified, and the type-3 cancellation (hence Vdet_q=Mdet_q) is not established for the multiparameter case claimed in Theorem 1.1. The q=1 and one-parameter q cases do not share this specific failure.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper claims a polynomial-size algebraic branching program (ABP) computing the q-Cayley determinant of q-right-quantum (q-RQ) matrices, including a multiparameter deformation, and hence, at q=-1, an exact polynomial-time ABP for the Cayley permanent of antisymmetric right-quantum matrices. The proof strategy is to prove Theorem 4.1, the equality Cdet_q(A)=Mdet_q(A)=Vdet_q(A) for q-RQ matrices, and then to compute the Valiant determinant by an explicit dynamic-programming ABP modeled on Mahajan–Vinay. The paper proves the Cartier–Foata case cleanly, gives an explicit O(n^3) ABP for the Valiant determinant in the free algebra, and then attempts to transfer the RQ/q-RQ equality through type-1, type-2, and type-3 collective cancellations of clow sequences.","tokens_in":29232,"tokens_out":15194,"duration_ms":161615,"significance":"If the main theorem is correct, it is a striking positive result: a noncommutative determinant of quantum-matrix type computable by a polynomial-size ABP, despite Nisan-type lower bounds for the free algebra, and an unusual exact permanent computation at q=-1. The Dynamic Programming Lemma 4.4 is explicit and appears sound, and the Cartier–Foata section gives a clean weight-preserving argument. There are no fitted parameters and no circular reliance on the main equality: the paper builds on the published parameter-free bijective result [KP07] for the CF/q-CF part, which is legitimate. However, the transfer of the RQ cancellation argument to the multiparameter q-RQ setting contains load-bearing gaps, and one step used repeatedly is false as stated. The full claim of Theorem 1.1 is therefore not established by the present proof.","major_comments":[{"comment":"The auxiliary matrix Q, obtained by replacing the first column of A by the k-th column, is not generally q-RQ for multiparameter q. The assertion that this is true because the columns of Q form a subcollection of columns of A ignores that the q_{ij} parameters in (3.2) depend on the column positions of Q, not on the original column labels. Concretely, take n=3, k=2, q_{12}=u, q_{13}=v, q_{23}=w with v≠w. Comparing the defining relation (3.2) for Q (with k=2, ℓ=3, i=1, j=3) with the corresponding relation for A yields (w-v)(a_{22}a_{33}-w a_{32}a_{23})=0 in B^q_n. No relation in the q-RQ ideal forces a_{22}a_{33}=w a_{32}a_{23} for generic A; at q=1 this would assert a commutativity that is not present. Thus Q∉B^q_n in general. Consequently the applications of Theorem 9.1 and Lemma 3.1 to Q in Lemma 9.6 are unjustified, and the type-3 collective cancellation, and hence Vdet_q=Mdet_q, is n","section":"§9.6, Lemma 9.6 and Eq. (3.2)"},{"comment":"The proof asserts that when a swap a_{ℓ j} a_{k i} is replaced by a_{k i} a_{ℓ j}, relations (3.2) give a_{ℓ j} a_{k i}=c a_{k i} a_{ℓ j} with one of the three scalar factors c. This is false in the q-RQ algebra for k≠ℓ: the second relation in (3.2) is a four-term identity, not a monomial equality. The q=1 RQ proof of Lemma 8.1 correctly uses a pairwise cancellation involving two partner q-sequences and the full four-term relation; the q-RQ proof as written instead treats the relation as a single-term swap. Therefore the claimed sign-preserving, weight-preserving involution Ψ is not established, and Cdet_q=Mdet_q in Lemma 4.2 needs a genuinely pairwise argument through (3.2).","section":"§9.1, proof of Theorem 9.1"},{"comment":"The reduction of the support to an interval {1,...,k,k,...,n} by relabelling larger elements is not an invariant operation for multiparameter q. The q-weights (3.10) contain products of q_{ij} attached to the actual labels, and a relabelling of support elements changes those scalars unless the parameter array is transformed as well. No such transformation is supplied. This WLOG reduction is used throughout the type-3 proof, including Lemma 9.6's normalization to head h=1, so the collective cancellation argument is not valid for generic multiparameter arrays as written.","section":"§8.4, Remark 8.13, applied in §9.2"},{"comment":"Independently of the Q∈B^q_n problem, the asserted sign-reversing, weight-matching bijection between type-3 q-clow sequences and cycle decompositions of Q is not proved. The text says the q-weights match 'as in the proof of Lemma 7.4', but Lemma 7.4 is a Cartier–Foata statement; in the q-RQ setting the bijection must track the q-inversion factors (7.1) across the deletion of the repeated vertex k. Because the q parameters for Q differ from those of the original clow-word positions, this matching is exactly where the multiparameter issue is most delicate, and the manuscript provides no calculation.","section":"§9.6, proof of Lemma 9.6"}],"minor_comments":[{"comment":"Typos: 'we proof that' should be 'we prove that'; the same verb form appears in a few places. This is purely editorial.","section":"§5 and §6"},{"comment":"Lemma 9.3 is proved before Lemma 9.6 is stated but invokes it for smaller support size. A forward reference or a reorganization of the induction would improve readability.","section":"§9.2"},{"comment":"The statement says 'polynomial time' while the ABP has coefficients involving the arbitrary nonzero complex numbers q_{ij}; the intended field/model of computation (e.g., algebraic computation over Q when q_{ij}∈Q) should be stated explicitly.","section":"Theorem 1.1"}],"recommendation":"major_revision","confidential_remarks":"The manuscript has a clean CF case and an explicit DP construction, but the advertised multiparameter Theorem 1.1 rests on a false auxiliary-matrix claim in Lemma 9.6 and on an unproved four-term swap argument in Theorem 9.1. The q=1 and one-parameter q cases may well be repairable, but the multiparameter extension as written is not established."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Quick take: the paper has a genuine and interesting result — a polynomial-size ABP for the Cayley determinant of right-quantum matrices — but the multiparameter q version advertised in the title and abstract is not established by the proof as written. The RQ (q=1) case and the single-parameter q case look solid; the multiparameter generalization is where the proof breaks.\n\nWhat is actually new: extending Mahajan–Vinay to compute a q-deformed Valiant determinant via dynamic programming, plus the combinatorial equalities Cdet = Mdet = Vdet. The CF case is clean and the Factorization Lemma is tidy. The q=1 RQ proof, building on Konvalinka–Pak, is plausible and gives the first positive complexity result of this kind, contrasting Nisan's lower bound. Credit where due.\n\nThe soft spots: Lemma 9.6 is the load-bearing one. The auxiliary matrix Q is formed by duplicating a column (or row, depending on convention). In the multiparameter q-RQ algebra, Q is not generally q-RQ: relations among the duplicated variables demand q_{1j}, but the original variables carry q_{kj}. Unless these coincide, applying Theorem 9.1 and Lemma 3.1 to Q is unjustified. The stress-test example with n=3 shows the two relations force a nontrivial identity that does not hold for generic variables. So the type-3 collective cancellation, and hence Vdet_q = Mdet_q for general q, is unproven.\n\nThere is also a gap in Theorem 9.1: the proof says swapping adjacent factors is a constant multiple in q-RQ, but (3.2) is not an individual swap identity; the correct argument is the pairwise cancellation used in Section 8. This is repairable in the one-parameter case, but as written it is not correct. Lemma 3.1 is cited, not proved; acceptable if the reference is exact, but for a paper whose theorem rests on it, a short proof or precise citation would help.\n\nProportionate bottom line: for q=1 and q scalar, the argument likely survives; the advertised multiparameter generality does not. This is a serious paper with a clear combinatorial core and a real payoff if the multiparameter issue can be patched, or if the claim is restricted. It deserves a full peer review, not a desk reject, but the referee should be asked to focus on Lemma 9.6 and Theorem 9.1. I would not cite the multiparameter theorem in its current form; I would consider citing the RQ/CF cases once the revision lands.","headline":"Multiparameter q extension is unproven as written; the RQ and one-parameter q cases look correct and are the real contribution.","tokens_in":29676,"tokens_out":8954,"would_cite":false,"duration_ms":79878,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["15A15","16T20","68Q25","05A05"],"pacs":[],"model":"deepseek-v4-flash","headline":"Quantum determinants can be computed by small algebraic branching programs.","keywords":["quantum determinant","right-quantum matrices","Cayley determinant","Moore determinant","Valiant determinant","algebraic branching program","noncommutative permanent","clow sequences"],"falsifier":"For n = 3 with generic parameters q12, q13, q23, write out the six type-3 clow sequences on {1,2,2,3}, compute their q-weighted signed sum T3 as in the paper's Example 8.19, and simplify using the q-RQ relations; if a single monomial in the entries survives, the collective cancellation fails. Alternatively, evaluate Cdet_q and Vdet_q on an explicit 4x4 q-RQ matrix with noncommuting entries and compare the two polynomials.","tokens_in":28707,"feed_emoji":"🧮","tokens_out":3792,"duration_ms":36531,"temperature":0.7,"pith_summary":"The paper aims to show that the Cayley determinant of a right-quantum (RQ) matrix—a noncommutative determinant whose entries obey quantum-group relations—can be computed by an algebraic branching program of polynomial (cubic) size, over the rationals and for every nonzero multiplicative parameter q. This is surprising because for the free noncommutative algebra the same polynomial requires exponential-size branching programs. The proof establishes a three-way equality between the Cayley, Moore, and Valiant determinants for q-RQ matrices, then adapts a standard dynamic-programming construction for the commutative determinant to compute the Valiant form. At q = -1 it yields an exact polynomial-time algorithm for the noncommutative permanent of antisymmetric right-quantum matrices—an unusual exact permanent computation.","feed_headline":"Polynomial-time algorithm computes quantum determinants","feed_subtitle":"A cubic-size branching program evaluates the Cayley determinant of right-quantum matrices, and at q=-1 the permanent too.","key_machinery":"The load-bearing objects are three noncommutative determinant forms: the Cayley determinant (sum over permutations in column order), the Moore determinant (sum over cycle decompositions), and the Valiant determinant (sum over all closed-walk 'clow' sequences). The argument uses two bijective/combinatorial transformations—a rank-increasing swap map on balanced words for the Cayley–Moore equality, and a type-by-type cancellation of non-cycle-decomposition clow sequences for the Moore–Valiant equality—together with a factorization lemma that reduces q-weights of clow sequences to products over single clows. The most delicate part is the type-3 cancellation, which pairs leftover terms with cycle","core_discovery":"The central discovery is Theorem 4.1: for any q-RQ matrix, the column-ordered Cayley determinant, the cycle-ordered Moore determinant, and the closed-walk (clow) Valiant determinant coincide as elements of the q-RQ algebra. This is not true in the free algebra; the equality exploits the commutation and two-by-two minor relations. Theorem 1.1 then follows by combining the equality with a polynomial-size branching program that evaluates the Valiant determinant directly from its clow-sequence definition, with the q-weights factored through a factorization lemma. Consequently, exact determinant and, at q = -1, exact permanent evaluation become tractable for these quantum matrix algebras.","pith_inferences":["If the type-3 cancellation can be formalized as a genuine weight-matching involution rather than a many-to-many correspondence, the same proof scheme might extend to generalized R-matrix algebras, a direction the authors suggest for Belavin–Drinfeld structures.","The exact permanent result at q = -1 is a natural stress test: implementing the ABP for small antisymmetric matrices and comparing its output with direct summation for n = 4 would expose any residual mismatch in the collective cancellation.","A concrete extension would be to replace the sign character in Cdet_q by other characters to obtain quantum immanants; the Cayley–Moore swap step would likely still work, but the Moore–Valiant cancellation would probably require new arguments."],"forward_implications":["Polynomial-time computation of Cayley determinants of RQ matrices is achieved, giving the first positive complexity result for quantum-matrix determinants.","At q = -1, the same ABP computes the noncommutative permanent of antisymmetric right-quantum matrices exactly in polynomial time.","The result extends to the multiparameter q-deformation, covering q-CF and q-RQ matrices uniformly.","The construction is division-free and of size O(n^3), so the algorithm is explicit and uniform over all nonzero parameter choices.","The equality Cdet = Mdet = Vdet isolates a structural property of RQ algebras that may hold for other R-matrix deformations."],"fun_headline_variants":["Quantum determinant falls to polynomial-time branching program","Cubic-size program computes quantum determinant","First efficient quantum determinant for quantum groups","Quantum permanent also tractable at q=-1"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The proof leans on the claim that the 'type-3' leftover clow sequences admit a sign-reversing bijection that matches their q-weights exactly with cycle decompositions of a q-RQ matrix with two equal columns, whose determinant is known to vanish.","fun_headline_variants_meta":{"raw":{"variants":["Quantum determinant falls to polynomial-time branching program","Cubic-size program computes quantum determinant","First efficient quantum determinant for quantum groups","Quantum permanent also tractable at q=-1"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000363,"raw_usage":{"total_tokens":1732,"prompt_tokens":618,"completion_tokens":1114,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":362,"completion_tokens_details":{"reasoning_tokens":1060}},"tokens_in":362,"tokens_out":1114,"duration_ms":10938,"temperature":1.0,"reasoning_tokens":1060,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-02T05:56:40.373687+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"For n = 3 with generic parameters q12, q13, q23, write out the six type-3 clow sequences on {1,2,2,3}, compute their q-weighted signed sum T3 as in the paper's Example 8.19, and simplify using the q-RQ relations; if a single monomial in the entries survives, the collective cancellation fails. Alternatively, evaluate Cdet_q and Vdet_q on an explicit 4x4 q-RQ matrix with noncommuting entries and compare the two polynomials.","supporting_citations":[],"review_version":1}